Asian Journal of Mathematics

Tautological module and intersection theory on Hilbert schemes of nodal curves

Ziv Ran

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Abstract

This paper presents the rudiments of Hilbert-Mumford Intersection (HMI) theory: intersection theory on the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension. We introduce an additive group of geometric cycles, called ’tautological module’, generated by diagonal loci, node scrolls, and twists thereof. We determine recursively the intersection action on this group by the discriminant (big diagonal) divisor and all its powers. We show that this suffices to determine arbitrary polynomials in Chern classes, in particular Chern numbers, for the tautological vector bundles on the Hilbert schemes, which are closely related to enumerative geometry of families of nodal curves.

Article information

Source
Asian J. Math. Volume 17, Number 2 (2013), 193-264.

Dates
First available in Project Euclid: 8 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1383923851

Mathematical Reviews number (MathSciNet)
MR3078931

Zentralblatt MATH identifier
1282.14097

Subjects
Primary: 14N99: None of the above, but in this section 14H99: None of the above, but in this section

Keywords
Hilbert scheme nodal curves intersection theory enumerative geometry

Citation

Ran, Ziv. Tautological module and intersection theory on Hilbert schemes of nodal curves. Asian J. Math. 17 (2013), no. 2, 193--264.https://projecteuclid.org/euclid.ajm/1383923851


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